Let me start with a simple observation. Suppose $f$ is a weight two newform of level $p^3$. Write $d$ for the size of the Galois orbit $f^\sigma, \sigma \in \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. Then $d \geq (p-1)/2$. The proof is quite simple: associated to $f$ is an abelian variety $A_f$ of dimension $d$ and conductor $p^{3d}$. If a prime $p$ divides the conductor of an abelian variety of dimension $d$, and $p>2d+1$, then the maximal power of $p$ dividing the conductor is $p^{2d}$ (I believe this is a theorem of Serre-Tate). Hence if we had $d < (p-1)/2$, this would immediately imply the absurd inequality $3d<2d$, so contradiction. Of course this also works for newforms of weight two and higher prime power level.

My questions:

Is this written down in the literature somewhere? It seems like such a simple observation that I want to guess that it is, but I cannot find a reference.

Is the same result true for higher weight newforms?